Performance Analysis of Convex Data Detection in MIMO

We study the performance of a convex data detection method in large multiple-input multiple-output (MIMO) systems. The goal is to recover an n-dimensional complex signal whose entries are from an arbitrary constellation $\mathcal{D} \subset \mathbb{C}$, using m noisy linear measurements. Since the Maximum Likelihood (ML) estimation involves minimizing a loss function over the discrete set ${\mathcal{D}^n}$, it becomes computationally intractable for large n. One approach is to relax to a $\mathcal{D}$ convex set and to utilize convex programing to solve the problem precise and then to map the answer to the closest point in the set $\mathcal{D}$. We assume an i.i.d. complex Gaussian channel matrix and derive expressions for the symbol error probability of the proposed convex method in the limit of m, n → ∞. Prior work was only able to do so for real valued constellations such as BPSK and PAM. The main contribution of this paper is to extend the results to complex valued constellations. In particular, we use our main theorem to calculate the performance of the complex algorithm for PSK and QAM constellations. In addition, we introduce a closed-form formula for the symbol error probability in the high-SNR regime and determine the minimum number of measurements m required for consistent signal recovery.